A proof of topological completeness for S4 in (0, 1)

The completeness of the modal logic S 4 for all topological spaces as well as for the real line R , the n -dimensional Euclidean space R n and the segment (0, 1) etc. (with □ interpreted as interior) was proved by McKinsey and Tarski in 1944. Several simplified proofs contain gaps. A new proof presented here combines the ideas published later by G. Mints and M. Aiello, J. van Benthem, G. Bezhanishvili with a further simplification. The proof strategy is to embed a finite rooted Kripke structure K for S 4 into a subspace of the Cantor space which in turn encodes (0, 1). This provides an open and continuous map from (0, 1) onto the topological space corresponding to K . The completeness follows as S4 is complete with respect to the class of all finite rooted Kripke structures.